{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Differentiable Pressure\n",
    "\n",
    "[![Google Collab Book](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/tum-pbs/PhiFlow/blob/develop/examples/optim/Differentiable_Pressure.ipynb)\n",
    "\n",
    "This example demonstrates how differentiating through the pressure allows us to find control parameters to approxmate a desired flow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# %pip install --quiet phiflow\n",
    "from phi.jax.flow import *\n",
    "# from phi.flow import *  # If JAX is not installed. You can use phi.torch or phi.tf as well.\n",
    "from tqdm.notebook import trange"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following setup: We can control the fluid flow inside two spherical regions on the left. The task is to approximate the desired `target` flow in the right half of the room."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 288x216 with 2 Axes>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 288x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "res = dict(x=80, y=64)\n",
    "control_area = union(Sphere(x=20, y=15, radius=10), Sphere(x=20, y=50, radius=10))\n",
    "control_mask = StaggeredGrid(control_area, 0, **res)\n",
    "target_mask = StaggeredGrid(Box(x=(40, INF), y=None), 0, **res)\n",
    "target = target_mask * StaggeredGrid(lambda x: math.exp(-0.5 * math.vec_squared(x - (50, 10), 'vector') / 32**2), 0, **res) * (0, 2)\n",
    "plot(control_mask['x'], target, overlay='args', size=(4, 3), title='Control areas (left) and target flow (right)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To optimize the control flow, we first define the loss function and get its derivative."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "@jit_compile\n",
    "def loss(v0):\n",
    "    v1, p = fluid.make_incompressible(v0 * control_mask)\n",
    "    return field.l2_loss((v1 - target) * target_mask), v1, p\n",
    "\n",
    "grad_fun = field.functional_gradient(loss, wrt='v0', get_output=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can iteratively apply gradient descent steps in order to reduce the loss."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "a92f05260a054fe781df440000d19dc5",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/20 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def gradient_descent_step(v0, _l, _v, step_size=1.):\n",
    "    (loss, v, p), dv0 = grad_fun(v0)\n",
    "    return v0 - step_size * dv0, loss, v\n",
    "\n",
    "velocity_fit = StaggeredGrid(Noise(), 0, **res) * 0.1 * control_mask\n",
    "ctrl_trj, loss_trj, v_trj = iterate(gradient_descent_step, batch(iter=20), velocity_fit, None, None, range=trange)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see how the loss is evolving."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 216x216 with 1 Axes>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 216x216 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(loss_trj.iter.as_spatial(), size=(3, 3), title=\"Loss\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can visualize how the full flow field changes during the optimization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x2017178abe0>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(control_area, v_trj, overlay='args', animate='iter', frame_time=200, size=(5, 4))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "phiflow2",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
